Merav Parter (Weizmann): Distributed Computing Made Secure: A New Cycle Cover Theorem

Abstract: 

In the area of distributed graph algorithms a number of network's entities with 

local views solve some computational task by exchanging messages with their 

neighbors. Quite unfortunately, an inherent property of most existing 

distributed algorithms is that throughout the course of their execution, the 

nodes get to learn not only their own output but rather learn quite a lot on 

the inputs or outputs of many other entities. This leakage of information might 

be a major obstacle in settings where the output (or input) of network's individual is a 

private information (e.g., distributed networks of selfish agents, 

decentralized digital currency such as Bitcoin).

 

While being quite an unfamiliar notion in the classical distributed setting,  

the notion of secure multi-party computation (MPC) is one of the main 

themes in the Cryptographic community. The existing secure MPC protocols do not quite fit the framework of classical distributed models in which only messages of bounded size are sent on graph edges in each round.

 

In this paper, we introduce a new framework for secure distributed graph and provide the first general compiler that takes any ``natural'' non-secure distributed algorithm that runs in $r$ rounds, and turns it into a secure algorithm that runs in $\widetilde{O}(r \cdot D \cdot 

\poly(\Delta))$ rounds where $\Delta$ is the maximum degree in the graph and $D$ is its diameter. 

A ``natural'' distributed algorithm is one where the local computation at each node can be performed in 

polynomial time. An interesting advantage of our approach is that it allows one to decouple between the price of locality and the price of security of a given graph function $f$. The security of the compiled algorithm is information-theoretic but holds only against a semi-honest adversary that 

controls a single node in the network. 

 

The main technical part of our compiler is based on a new cycle cover theorem: 

We show that the edges of every bridgeless graph $G$ of diameter $D$ can be 

covered by a collection of cycles such that each cycle is of length 

$\widetilde{O}(D)$ and each edge of the graph $G$ appears in 

$\widetilde{O}(1)$ many cycles, existentially this is optimal, upto polylogarithmic terms.